On the coefficient-choosing game

TL;DR

This paper studies a polynomial coefficient game over finite cyclic rings, extending previous theories, and constructs examples of polynomials with specific root properties using advanced algebraic techniques.

Contribution

It generalizes the coefficient-choosing game to all finite cyclic rings and introduces methods to construct polynomials with prescribed root characteristics.

Findings

Extended the theory to all finite cyclic rings.

Constructed polynomials with no roots in certain algebraic extensions.

Developed an adversarial approach for polynomial construction.

Abstract

Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to $3$ or greater than $8$) with no roots in the maximal abelian extension of $\mathbb{Q}$.